Well, the font named Symbol is indeed a symbol font. There are many fonts that have been created that do not contain letters of any alphabet. I exclude special fonts used for the display of other languages; e.g., Chinese, Japanese, Korean, Hebrew, Arabic.

Symbol fonts are called ornaments, decorations, dingbats, pi, and sorts. The last two terms both refer to the fact that these are unordered, or out of the ordinary. They come from the old cold lead type days, when after a page was printed, the individual letters that comprised it were broken down, restored to their cases, and stored for reuse. Symbols, few in number and not belonging to the font used to set the rest of the page, were thrown into a box to be sorted out later by an apprentice.

I would dearly love to demonstrate some symbol fonts for you, but I can't think of any way of doing so that won't break on some machines, especially here on E2. I'd be happy to post a GIF of a few, but that's forbidden.

In the old days, you could not work with type without resort to a typebook, a great binder containing samples of every type font in the house. In the modern age, it's easy to choose a font on the screen from a menu, see how it looks, and change it if you don't like it. But a typebook is still a useful tool, especially when dealing with symbol fonts. It can be hard to find the special character you want, since it will generally be bound to a keypress that has nothing to do with its meaning -- if any.

Tools to help you with symbol fonts include (on Macintosh) Key Caps, Adobe Type Reunion, and Adobe Type Manager. And while I hesitate to endorse any of Devil Gates' creations, Microsoft Word has a very useful feature. Choose "Symbol" from the "Insert" menu, and you get a table of every character in the current font. You can continue to type, or use your mouse. Either way, you get the needed keypress read out on screen. Before opening the Symbol table, choose the font you want and a fairly large size.